pivoting, the procedure is known as full pivoting.
When exchanging rows and columns in partial or full pivoting, it is necessary
to keep track of the exchanges because the order of the unknowns in the
solution is altered by those exchanges.
exchanges in partial or full pivoting mode, is to create a permutation matrix P
, at the beginning of the procedure.
required in the augmented matrix A
exchange, respectively, in the permutation matrix.
achieved, then, we multiply the permutation matrix by the unknown vector x
to obtain the order of the unknowns in the solution. In other words, the final
solution is given by P⋅x = b', where b' is the last column of the augmented
matrix after the solution has been found.
Example of Gauss-Jordan elimination with full pivoting
Let's illustrate full pivoting with an example. Solve the following system of
equations using full pivoting and the Gauss-Jordan elimination procedure:
The augmented matrix and the permutation matrix are as follows:
Store the augmented matrix in variable AAUG, then press ‚ @AAUG to get a
copy in the stack. We want to keep the CSWP (Column Swap) command
readily available, for which we use: ‚N~~cs~ (find CSWP),
@@OK@@. You'll get an error message, press $, and ignore the message.
Next, get the ROW menu available by pressing: „Ø @) C REAT @) @ ROW@.
Now we are ready to start the Gauss-Jordan elimination with full pivoting.
We will need to keep track of the permutation matrix by hand, so take your
notebook and write the P matrix shown above.
When row and column exchanges are allowed in
One way to keep track of column
is also registered as a row or column
X + 2Y + 3Z = 2,
3Z = -1,
8X +16Y- Z = 41.
Any row or column exchange
When the solution is